7.2 Wave Types

The ocean supports a remarkable diversity of wave phenomena spanning wavelengths from millimeters (capillary waves) to thousands of kilometers (planetary Rossby waves). Each wave type is governed by distinct restoring forces and exhibits unique dispersion characteristics.

Capillary Waves

When wavelengths are shorter than approximately 1.7 cm, surface tension (γ) becomes the dominant restoring force rather than gravity. The full dispersion relation including both gravity and surface tension is:

$$\omega^2 = \left(gk + \frac{\gamma}{\rho}k^3\right)\tanh(kh)$$

where γ ≈ 0.074 N/m for seawater at 20°C

The minimum phase speed occurs at the capillary-gravity transition wavelength:

$$\lambda_{\min} = 2\pi\sqrt{\frac{\gamma}{\rho g}} \approx 1.7 \text{ cm}, \quad c_{\min} = \left(\frac{4g\gamma}{\rho}\right)^{1/4} \approx 0.23 \text{ m/s}$$

Derivation: Capillary-Gravity Minimum Phase Speed

Step 1. In the deep water limit ($\tanh(kh) \to 1$), the full dispersion relation reduces to:

$$\omega^2 = gk + \frac{\gamma}{\rho}k^3$$

Step 2. The phase speed is $c = \omega/k$, so $c^2 = g/k + \gamma k/\rho$. The gravity term $g/k$ dominates at small k (long waves) while the capillary term $\gamma k/\rho$ dominates at large k (short waves), so a minimum exists.

Step 3. Minimize $c^2$ by setting $d(c^2)/dk = 0$:

$$\frac{d(c^2)}{dk} = -\frac{g}{k^2} + \frac{\gamma}{\rho} = 0 \quad \Longrightarrow \quad k_{\min} = \sqrt{\frac{\rho g}{\gamma}}$$

Step 4. The corresponding wavelength and minimum phase speed are:

$$\lambda_{\min} = \frac{2\pi}{k_{\min}} = 2\pi\sqrt{\frac{\gamma}{\rho g}}, \quad c_{\min}^2 = \frac{g}{k_{\min}} + \frac{\gamma k_{\min}}{\rho} = 2\sqrt{\frac{g\gamma}{\rho}}$$

$$\boxed{c_{\min} = \left(\frac{4g\gamma}{\rho}\right)^{1/4} \approx 0.23 \text{ m/s}, \quad \lambda_{\min} \approx 1.7 \text{ cm}}$$

Waves with phase speeds below $c_{\min}$ cannot propagate, which is why wind below ~0.23 m/s cannot generate surface waves. At $k = k_{\min}$, the gravity and capillary contributions to the phase speed are exactly equal.

Properties

Capillary waves are anomalously dispersive: shorter waves travel faster (opposite to gravity waves). They are rapidly damped by viscosity.

Importance

Capillary waves roughen the sea surface, affecting radar backscatter (SAR imagery) and gas exchange rates across the air-sea interface.

Gravity Waves and Swell Propagation

Wind-generated gravity waves with periods of 1–25 seconds dominate the ocean surface. Once they leave the generation area, they propagate as swell across entire ocean basins with minimal energy loss:

$$c_g = \frac{gT}{4\pi} \quad \text{(deep water group velocity)}$$

A 15-second swell has $c_g \approx 11.7$ m/s and can cross the Pacific (~10,000 km) in about 10 days. During propagation, angular spreading and frequency dispersion cause the spectrum to narrow, producing clean, long-crested swell. Swell from Southern Ocean storms has been tracked propagating more than 15,000 km to reach the Alaskan coast.

Sea vs Swell

Sea state refers to locally wind-generated waves with broad directional and frequency spectra (short-crested, chaotic). Swell is the far-field propagated component with narrow spectra (long-crested, regular). The Pierson-Moskowitz (PM) spectrum describes fully-developed sea, while swell is better described by a Gaussian spectral peak.

Wave Age

The wave age $c_p/U_{10}$ (ratio of peak phase speed to wind speed) distinguishes young wind seas ($c_p/U_{10} < 1$) from mature seas ($c_p/U_{10} \approx 1.2$). Young seas are steeper and contribute more to air-sea drag. Old swell ($c_p/U_{10} \gg 1$) can actually reduce surface roughness.

Infragravity Waves (25–300 s)

Generated by nonlinear interactions between wind wave groups (bound long waves released upon breaking). Infragravity waves dominate surf zone motions, drive harbor resonance (seiching), and contribute to microseismic noise recorded by seismometers worldwide.

Seiches (Standing Waves in Enclosed Basins)

Seiches are standing waves in enclosed or semi-enclosed water bodies (lakes, harbors, bays). Merian's formula gives the natural period for the fundamental mode:

$$T_n = \frac{2L}{n\sqrt{gh}}, \quad n = 1, 2, 3, \ldots$$

L is the basin length, h is the mean depth, n is the mode number

Seiches can be triggered by wind stress changes, atmospheric pressure gradients, seismic events, or tsunamis. Lake Geneva's seiche has a period of about 73 minutes. Harbor seiches can amplify wave energy at resonant frequencies, causing significant damage to moored vessels.

Internal Waves

Internal waves propagate along density interfaces within the ocean interior. They can have amplitudes exceeding 100 m while producing only centimeter-scale surface signatures. The dispersion relation for internal waves in a continuously stratified fluid is:

$$\omega^2 = N^2\frac{k_h^2}{k_h^2 + m^2} = N^2\cos^2\theta$$

N is the buoyancy (Brunt-Väisälä) frequency, $k_h$ is horizontal wavenumber, m is vertical wavenumber, θ is the angle from horizontal

Derivation: Internal Wave Dispersion Relation

Step 1. Start from the linearized Boussinesq equations for an incompressible, stratified fluid at rest. The momentum and buoyancy equations are:

$$\frac{\partial u}{\partial t} = -\frac{1}{\rho_0}\frac{\partial p'}{\partial x}, \quad \frac{\partial w}{\partial t} = -\frac{1}{\rho_0}\frac{\partial p'}{\partial z} + b, \quad \frac{\partial b}{\partial t} = -N^2 w$$

where the buoyancy $b = -g\rho'/\rho_0$ and $N^2 = -(g/\rho_0)(d\bar{\rho}/dz)$ is the Brunt-Väisälä frequency squared.

Step 2. Assume plane wave solutions of the form $[u, w, p', b] \propto \exp\bigl(i(k_h x + m z - \omega t)\bigr)$. Substituting into the equations converts time and spatial derivatives to algebraic factors:

$$-i\omega \hat{u} = -\frac{ik_h}{\rho_0}\hat{p}, \quad -i\omega \hat{w} = -\frac{im}{\rho_0}\hat{p} + \hat{b}, \quad -i\omega \hat{b} = -N^2 \hat{w}$$

Step 3. From the buoyancy equation, $\hat{b} = N^2 \hat{w} / (i\omega)$. Substituting into the vertical momentum equation and combining with continuity $k_h \hat{u} + m \hat{w} = 0$, eliminate $\hat{u}$ and $\hat{p}$:

$$\omega^2(k_h^2 + m^2) = N^2 k_h^2$$

Step 4. Solving for $\omega^2$ and noting that $k_h / \sqrt{k_h^2 + m^2} = \cos\theta$ where $\theta$ is the angle of the wave vector from the horizontal:

$$\boxed{\omega^2 = N^2 \frac{k_h^2}{k_h^2 + m^2} = N^2 \cos^2\theta}$$

This shows that internal wave frequency depends only on the angle of propagation, not the magnitude of the wave vector. The frequency is bounded: $0 \leq \omega \leq N$, with $\omega = N$ for horizontal propagation ($\theta = 0$) and $\omega = 0$ for vertical propagation.

For a two-layer ocean with densities $\rho_1$ and $\rho_2$ and layer thicknesses $h_1$ and $h_2$, the interfacial wave speed is:

$$c_i = \sqrt{\frac{g(\rho_2 - \rho_1)}{\rho_2}\frac{h_1 h_2}{h_1 + h_2}}$$

Internal waves are generated by tidal flow over topography (internal tides), wind forcing, and flow instabilities. They play a critical role in diapycnal mixing that maintains the ocean's stratification.

Kelvin Waves

Kelvin waves are a special class of gravity waves that propagate along boundaries (coastlines or the equator) with the boundary on the right in the Northern Hemisphere (left in the Southern). The cross-boundary velocity is exactly zero:

$$\eta = \eta_0 \exp\left(-\frac{y}{R_d}\right)\cos(kx - \omega t)$$

where $R_d = \sqrt{gH}/f$ is the Rossby radius of deformation

Derivation: Coastal Kelvin Wave Solution

Step 1. Begin with the linearized shallow water equations on an f-plane, with x along the coast and y offshore:

$$\frac{\partial u}{\partial t} - fv = -g\frac{\partial \eta}{\partial x}, \quad \frac{\partial v}{\partial t} + fu = -g\frac{\partial \eta}{\partial y}, \quad \frac{\partial \eta}{\partial t} + H\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}\right) = 0$$

Step 2. The key Kelvin wave ansatz is $v = 0$ everywhere (no cross-shore flow). This simplifies the equations to:

$$\frac{\partial u}{\partial t} = -g\frac{\partial \eta}{\partial x}, \quad fu = -g\frac{\partial \eta}{\partial y}, \quad \frac{\partial \eta}{\partial t} + H\frac{\partial u}{\partial x} = 0$$

Step 3. The first and third equations form a 1-D wave equation with solutions propagating at $c = \sqrt{gH}$. Try a separable solution $\eta = \hat{\eta}(y)\cos(kx - \omega t)$ with $\omega = ck$. Then $u = (g/c)\hat{\eta}(y)\cos(kx - \omega t)$.

Step 4. Substitute into the geostrophic balance $fu = -g\,\partial\eta/\partial y$:

$$\frac{f}{c}\hat{\eta} = -\frac{d\hat{\eta}}{dy} \quad \Longrightarrow \quad \hat{\eta}(y) = \eta_0 \exp\!\left(-\frac{y}{R_d}\right), \quad R_d = \frac{c}{f} = \frac{\sqrt{gH}}{f}$$

Step 5. The complete Kelvin wave solution is therefore:

$$\boxed{\eta = \eta_0 \exp\!\left(-\frac{y}{R_d}\right)\cos(kx - \omega t), \quad c = \sqrt{gH}, \quad v = 0}$$

The wave is non-dispersive and trapped within a Rossby radius of the coast. In the Northern Hemisphere ($f > 0$), the solution decays offshore only if the coast is on the right of the propagation direction, explaining why coastal Kelvin waves propagate with the coast on their right.

Coastal Kelvin Waves

Trapped to the coast with e-folding scale R_d (~30–100 km at mid-latitudes). Speed $c = \sqrt{gH}$. Important for tidal propagation and storm surge dynamics.

Equatorial Kelvin Waves

Trapped to the equator (R_d ~ 250 km). Propagate eastward only at $c \approx 2.8$ m/s for the first baroclinic mode. Central to ENSO dynamics.

Rossby (Planetary) Waves

Rossby waves arise from the conservation of potential vorticity on a rotating sphere, with the variation of the Coriolis parameter (β = df/dy) providing the restoring mechanism:

$$\omega = \frac{-\beta k}{k^2 + l^2 + R_d^{-2}}$$

β ≈ 2×10²¹³ m²¹s²¹, R_d is the Rossby deformation radius

Derivation: Rossby Wave Dispersion Relation

Step 1. Start from the barotropic vorticity equation obtained by taking the curl of the shallow water momentum equations. On a $\beta$-plane ($f = f_0 + \beta y$), conservation of potential vorticity linearized about a state of rest gives:

$$\frac{\partial}{\partial t}\left(\nabla^2 \psi - R_d^{-2}\psi\right) + \beta \frac{\partial \psi}{\partial x} = 0$$

where $\psi$ is the streamfunction ($u = -\partial\psi/\partial y$, $v = \partial\psi/\partial x$) and $R_d^{-2} = f_0^2/(gH)$ is the stretching term from free surface deformation.

Step 2. Assume a plane wave solution $\psi = \hat{\psi}\exp\bigl(i(kx + ly - \omega t)\bigr)$. Then $\nabla^2\psi = -(k^2+l^2)\psi$, and time/space derivatives become algebraic:

$$-i\omega\bigl(-(k^2 + l^2) - R_d^{-2}\bigr)\hat{\psi} + i\beta k\,\hat{\psi} = 0$$

Step 3. Since $\hat{\psi} \neq 0$, divide through and solve for $\omega$:

$$\boxed{\omega = \frac{-\beta k}{k^2 + l^2 + R_d^{-2}}}$$

Step 4. The zonal phase speed is $c_x = \omega/k = -\beta/(k^2 + l^2 + R_d^{-2})$. Since $\beta > 0$ and the denominator is always positive, $c_x < 0$ (always westward). Physically, the $\beta$-effect causes northward-displaced fluid to acquire negative relative vorticity and southward-displaced fluid to acquire positive relative vorticity, producing a westward phase propagation.

The phase speed is always westward:

$$c_x = \frac{\omega}{k} = \frac{-\beta}{k^2 + l^2 + R_d^{-2}}$$

Barotropic Rossby Waves

Involve entire water column. Very fast (tens of m/s). Important for ocean spin-up response to wind changes.

Baroclinic Rossby Waves

First-mode speed: $c \approx \beta R_d^2$, only ~2–5 cm/s at mid-latitudes. Takes years to cross ocean basins. Visible in satellite altimetry.

Edge Waves

Edge waves are trapped to the shoreline by refraction over a sloping beach. They propagate along the shore and decay exponentially offshore. For a planar beach with slope $\beta_b$:

$$\omega^2 = gk\sin\left[(2n+1)\beta_b\right], \quad n = 0, 1, 2, \ldots$$

Edge waves contribute to rhythmic beach morphology (cusps), rip current spacing, and crescentic bar patterns. They are excited by incoming wave groups and can persist as free modes after storm events.

Topographic Rossby Waves and Continental Shelf Waves

When the ocean floor is not flat, the conservation of potential vorticity introduces a topographic β-effect that supports topographic Rossby waves. These waves propagate with the shallow water on their right (Northern Hemisphere):

$$\omega = \frac{-f_0 \alpha k}{k^2 + l^2 + f_0^2/(gH)}$$

where $\alpha = (1/H)(dH/dy)$ is the bottom slope parameter, analogous to β for planetary Rossby waves

Continental Shelf Waves

Long-period (days) waves propagating along continental shelves. Excited by wind and pressure forcing. Periods of 3–20 days. Responsible for much of the sub-tidal sea level variability at the coast.

Topographic Trapping

Strong bottom slopes trap wave energy along features like mid-ocean ridges and seamount chains. The topographic β-effect can be 10–100 times larger than the planetary β-effect on continental slopes.

Python: Wave Type Dispersion & Kelvin Wave Animation

Python

!/usr/bin/env python3

script.py71 lines

#!/usr/bin/env python3
"""wave_types.py - Dispersion for capillary/gravity/internal/Rossby waves"""
import numpy as np
import matplotlib.pyplot as plt

g = 9.81
rho = 1025.0         # seawater density (kg/m^3)
gamma = 0.074        # surface tension (N/m)
f0 = 1e-4            # Coriolis parameter (1/s)
beta = 2e-11         # beta parameter (1/m/s)
N = 0.01             # buoyancy frequency (1/s)

# --- 1. Capillary-gravity wave phase speed vs wavelength ---
lam = np.logspace(-3, 3, 1000)  # wavelength in meters
k = 2 * np.pi / lam
omega_cg = np.sqrt(g * k + (gamma / rho) * k**3)  # deep water
c_cg = omega_cg / k

fig, axes = plt.subplots(2, 2, figsize=(14, 10))

ax1 = axes[0, 0]
ax1.loglog(lam, c_cg, 'b-', lw=2, label='Capillary-gravity (deep)')
ax1.axhline(np.sqrt(g * 10.0), color='r', ls='--',
            label='Shallow water c=sqrt(gh), h=10m')
ax1.axvline(0.017, color='gray', ls=':', label='lambda_min = 1.7 cm')
ax1.set_xlabel('Wavelength (m)'); ax1.set_ylabel('Phase speed (m/s)')
ax1.set_title('Phase Speed: Capillary-Gravity Transition')
ax1.legend(fontsize=8); ax1.grid(True, alpha=0.3)

# --- 2. Internal wave dispersion ---
theta = np.linspace(0.01, np.pi/2 - 0.01, 200)
omega_iw = N * np.cos(theta)
ax2 = axes[0, 1]
ax2.plot(np.degrees(theta), omega_iw * 1000, 'c-', lw=2)
ax2.axhline(f0 * 1000, color='r', ls='--', label=f'f = {f0} s^-1')
ax2.axhline(N * 1000, color='orange', ls='--', label=f'N = {N} s^-1')
ax2.set_xlabel('Angle from horizontal (degrees)')
ax2.set_ylabel('Frequency (10^-3 s^-1)')
ax2.set_title('Internal Wave Dispersion')
ax2.legend(); ax2.grid(True, alpha=0.3)

# --- 3. Rossby wave dispersion ---
k_r = np.linspace(1e-6, 5e-5, 200)
Rd = 50e3  # Rossby radius 50 km
omega_rossby = -beta * k_r / (k_r**2 + 1/Rd**2)
c_rossby = omega_rossby / k_r

ax3 = axes[1, 0]
ax3.plot(k_r * 1e5, -c_rossby * 100, 'purple', lw=2)
ax3.set_xlabel('Wavenumber (10^-5 rad/m)')
ax3.set_ylabel('Westward phase speed (cm/s)')
ax3.set_title('Baroclinic Rossby Wave Phase Speed')
ax3.grid(True, alpha=0.3)

# --- 4. Kelvin wave structure ---
Rd_kelvin = 100e3  # 100 km deformation radius
y = np.linspace(0, 400e3, 200)
x = np.linspace(0, 1000e3, 300)
X, Y = np.meshgrid(x, y)
k_kw = 2 * np.pi / 200e3
eta_kw = np.exp(-Y / Rd_kelvin) * np.cos(k_kw * X)

ax4 = axes[1, 1]
cf = ax4.contourf(X/1e3, Y/1e3, eta_kw, levels=20, cmap='RdBu_r')
plt.colorbar(cf, ax=ax4, label='Surface elevation')
ax4.set_xlabel('Alongshore x (km)'); ax4.set_ylabel('Offshore y (km)')
ax4.set_title('Coastal Kelvin Wave Structure')

plt.tight_layout()
plt.savefig('wave_types.png', dpi=150)
plt.show()

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Internal Wave Ray Tracing

Fortran

Ray tracing for internal waves in a stratified ocean

program.f9082 lines

program internal_wave_rays
  ! Ray tracing for internal waves in a stratified ocean
  ! with depth-varying buoyancy frequency N(z)
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.141592653589793_dp
  real(dp) :: x, z, kx, kz, omega, N_local, dNdz
  real(dp) :: dt_ray, cx, cz, dkx_dt, dkz_dt
  real(dp) :: N0, N_deep, z_pyc, dz_pyc
  integer :: i, nsteps

! N(z) profile: exponential with pycnocline enhancement
  N0 = 0.005_dp       ! surface N (rad/s)
  N_deep = 0.001_dp   ! deep N (rad/s)
  z_pyc = -200.0_dp   ! pycnocline depth (m)
  dz_pyc = 50.0_dp    ! pycnocline thickness (m)

! Initial conditions
  omega = 0.003_dp    ! wave frequency (between f and N)
  x = 0.0_dp
  z = -100.0_dp       ! starting depth
  kx = 2.0_dp * pi / 5000.0_dp  ! horizontal wavenumber
  ! Compute kz from dispersion: omega^2 = N^2 * kx^2/(kx^2+kz^2)
  N_local = buoyancy_freq(z, N0, N_deep, z_pyc, dz_pyc)
  kz = kx * sqrt(N_local**2 / omega**2 - 1.0_dp)

dt_ray = 60.0_dp    ! ray time step (s)
  nsteps = 10000

N_local = buoyancy_freq(z, N0, N_deep, z_pyc, dz_pyc)
  print '(A)', 'Step  x(km)  z(m)  N(s^-1)  kx  kz'
  print '(I6, F12.3, F12.1, F12.6, 2E14.4)', &
        0, x/1000.0_dp, z, N_local, kx, kz

do i = 1, nsteps
    N_local = buoyancy_freq(z, N0, N_deep, z_pyc, dz_pyc)
    dNdz = buoyancy_freq_deriv(z, N0, N_deep, z_pyc, dz_pyc)

! Group velocity components
    cx = N_local * kz**2 / (kx**2 + kz**2)**1.5_dp
    cz = -N_local * kx * kz / (kx**2 + kz**2)**1.5_dp

! Wavenumber evolution (dkx/dt = 0 for horiz uniform medium)
    dkx_dt = 0.0_dp
    dkz_dt = -kx * dNdz * kz / (kx**2 + kz**2)**0.5_dp

! Update position and wavenumber
    x = x + cx * dt_ray
    z = z + cz * dt_ray
    kx = kx + dkx_dt * dt_ray
    kz = kz + dkz_dt * dt_ray

! Check boundaries
    if (z > 0.0_dp .or. z < -4000.0_dp) then
      print '(A,I6,A,F8.1,A)', 'Ray exited domain at step ', i, &
            ', z = ', z, ' m'
      exit
    end if

if (mod(i, 20) == 0) then
      print '(I6, F12.3, F12.1, F12.6, 2E14.4)', &
            i, x/1000.0_dp, z, N_local, kx, kz
    end if
  end do

contains
  function buoyancy_freq(zz, n0, nd, zp, dzp) result(nn)
    real(dp), intent(in) :: zz, n0, nd, zp, dzp
    real(dp) :: nn
    nn = nd + (n0 - nd) * exp(zz / 500.0_dp) + &
         0.01_dp * exp(-((zz - zp) / dzp)**2)
  end function

function buoyancy_freq_deriv(zz, n0, nd, zp, dzp) result(dnn)
    real(dp), intent(in) :: zz, n0, nd, zp, dzp
    real(dp) :: dnn
    dnn = (n0 - nd) / 500.0_dp * exp(zz / 500.0_dp) + &
          0.01_dp * (-2.0_dp * (zz - zp) / dzp**2) * &
          exp(-((zz - zp) / dzp)**2)
  end function

end program internal_wave_rays

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Rogue Waves

Rogue (freak) waves are individual waves with height exceeding twice the significant wave height of the surrounding sea state:

$$H_{\text{rogue}} \geq 2 H_s \quad \text{or} \quad \text{abnormality index} = H_{\max}/H_s \geq 2$$

Once dismissed as sailors' tales, rogue waves were confirmed by the Draupner platform measurement on January 1, 1995 (25.6 m wave in a 12 m sea state). Generation mechanisms include:

Linear Superposition

Constructive interference of many wave components (Gaussian statistics predict ~1 in 3,000 waves exceeds 2H_s). Enhanced by directional focusing.

Nonlinear Focusing

Benjamin-Feir (modulational) instability causes energy concentration in narrow-banded seas. Described by the Nonlinear Schrodinger equation. Occurs when BFI > 1.

Wave-Current Interaction

Opposing currents (e.g., Agulhas Current) focus wave energy, dramatically increasing wave heights. Well-documented hazard off South Africa.

Crossing Sea States

Two wave systems meeting at oblique angles can create exceptionally steep and dangerous waves through constructive interference patterns.

The Ocean Wave Energy Spectrum

The full spectrum of ocean waves spans more than 10 orders of magnitude in frequency. The relative energy in each band follows from the restoring forces and generation mechanisms:

High-Frequency Band

Capillary and ultra-gravity waves (f > 1 Hz). Restoring force transitions from gravity to surface tension at $\lambda \approx 1.7$ cm. Rapidly damped by viscosity. Important for microwave remote sensing (scatterometry, SAR).

Gravity Wave Band

Wind waves and swell (0.03–1 Hz). Maximum ocean wave energy. Gravity is the restoring force. Dispersive in deep water, non-dispersive in shallow water. Carries most of the surface wave energy.

Low-Frequency Band

Infragravity, seiches, tsunamis (periods minutes to hours). Coriolis force becomes relevant. Long waves behave as shallow water waves. Generated by earthquakes, atmospheric pressure, and nonlinear wave-wave interactions.

Planetary Wave Band

Tides, Kelvin waves, Rossby waves (periods days to years). Rotation and β-effect provide restoring forces. Kelvin wave speed $c = \sqrt{gH}$ = 2–200 m/s. Rossby waves only 1–5 cm/s at mid-latitudes.

Equatorial Trapped Waves: Complete Spectrum

The equatorial ocean supports a rich spectrum of trapped waves due to the vanishing of the Coriolis parameter at the equator. These waves are solutions to the equatorial β-plane shallow water equations and are trapped within a few degrees of latitude with an e-folding scale equal to the equatorial Rossby radius:

$$R_{eq} = \sqrt{\frac{c}{\beta}} \approx 250 \text{ km (first baroclinic mode)}$$

where c is the internal gravity wave speed and $\beta = 2.3 \times 10^{-11}$ m²¹s²¹

The complete equatorial wave spectrum includes:

Equatorial Kelvin Waves

Eastward only, non-dispersive, c ≈ 2.8 m/s. Symmetric about equator. Zero cross-equatorial velocity. Crosses the Pacific in ~2 months. Triggers downwelling and thermocline deepening.

Equatorial Rossby Waves

Westward, dispersive, $c = -c_K/(2n+1)$. Mode 1 takes ~6 months to cross the Pacific. Reflect from western boundary as Kelvin waves. Central to ENSO dynamics.

Mixed Rossby-Gravity (Yanai) Waves

Transition between Rossby and gravity wave regimes. Antisymmetric about the equator. Important for generating tropical instability waves that transport heat meridionally.

Inertia-Gravity Waves

Propagate both east and west at speeds faster than the Kelvin wave. Higher meridional mode structures. Can be generated by westerly wind bursts during ENSO onset.

Wave Classification Summary

Surface Waves

Capillary (λ < 1.7 cm), gravity (1 cm–300 m), infragravity (km-scale). Restoring forces: surface tension + gravity.

Internal Waves

Periods between f and N. Amplitudes >100 m. Critical for deep ocean mixing. Generated at topography.

Planetary Waves

Kelvin: boundary-trapped, fast. Rossby: always westward, slow (cm/s). Both essential for ocean adjustment and ENSO.

← 7.1 Wave Dynamics 7.3 Tidal Forces →